Set \(f_n:=0\)  for all \(n<0\) . Then, (REF ) holds for every \(x, y\in \mathbb {R}^d\)  and \(n\in \mathbb {Z}\) . Choose \(c_1, \ldots , c_{r+1} \in \mathbb {R}\)  such that
\(0<c_1<a_1\le b_1<c_2 <a_2\le b_2 <\ldots a_r \le b_r <c_{r+1}.\) 
For each \(i\in \lbrace 1, 2, \ldots , r+1\rbrace \) , set
\(S_i:=\bigg \lbrace  x\in \mathbb {R}^d: \sup _{n\ge 0} \frac{1}{c_i^n} \Vert  \mathcal {B}(n,0)x\Vert <+\infty \bigg \rbrace .\) 
Then (see the proof of Proposition REF ), for each \(i\in \lbrace 1,2 \ldots , r+1\rbrace \) , the sequence \((\frac{1}{c_i}B_n)_{n\in \mathbb {Z}^+}\)  admits a nonuniform exponential dichotomy with respect to projections \(\tilde{P}_n^i\)  for \(n\in \mathbb {Z}^+\) , where \(\operatorname{Im}\tilde{P}_0^i=S_i\) . Moreover, the following holds true:

\(S_i\)  does not depend on the particular choice of \(c_i\) ;

\(S_1=\lbrace 0\rbrace  \subsetneq S_2\ldots \subsetneq S_r \subsetneq S_{r+1}=\mathbb {R}^d\) .

Next, choose subspaces \(V_2, \ldots , V_{r}\)  of \(\mathbb {R}^d\)  such that
\(\mathbb {R}^d=S_r\oplus V_r, \ S_r=S_{r-1} \oplus V_{r-1}, \ldots , S_3=S_2\oplus V_2.\) 
Hence,
\(\mathbb {R}^d=V_r\oplus V_{r-1} \oplus \ldots \oplus V_2 \oplus S_2.\) 
Define an operator \(B\)  on \(\mathbb {R}^d\)  by
\(B\vert _{S_2}=a_1\text{\rm Id}_{S_2}, \ B\vert _{V_2}=a_2\text{\rm Id}_{V_2}, \ldots , B\vert _{V_r}=a_r\text{\rm Id}_{V_r},\) 

where \(\text{\rm Id}_V\)  denotes the identity map on a subspace \(V\subset \mathbb {R}^d\) . Moreover,
set \(B_n:=B\)  and \(\Vert \cdot \Vert _n=\Vert  \cdot \Vert \)  for \(n<0\) .
Set \(\mathcal {S}^{\prime }:=\lbrace  \Vert \cdot \Vert _n; \ n\in \mathbb {Z}\rbrace \) , \(\mathbb {B}^{\prime }:=(B_n)_{n\in \mathbb {Z}}\) , and let \(\Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}\)  be defined as in the Definition REF .
We now claim that \(\Sigma _{ED, \mathbb {B}, \mathcal {S}}=\Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}\) . As mentioned above, \(S_i\)  does not depend on \(c_i\) . Thus, for any
\(\tilde{c}_1\in (0, a_1), ~~\tilde{c}_i\in (b_{i-1}, a_{i}), ~~ i\in \lbrace 2,...,r\rbrace , ~~ \tilde{c}_{r+1}\in (b_r, \infty ),\) 
we have that the sequence \((\frac{1}{\tilde{c}_i}B_n)_{n\in \mathbb {Z}^+}\)  also admits a strong exponential dichotomy with respect to the sequence of norms \(\mathcal {S}\)  and projections \(\tilde{P}_n^i\)  for all \(n\in \mathbb {Z}^+\)  such that
\(\operatorname{Im}\tilde{P}_0^i=S_{i}\) . Moreover, it is easy to see from (REF ) that the sequence \((\frac{1}{\tilde{c}_i}B_n)_{n\in \mathbb {Z}^-}\)  admits a strong exponential dichotomy on \(\mathbb {Z}^-\)  with respect to the sequence of norms \(\Vert  \cdot \Vert _n\) , \(n\in \mathbb {Z}^-\)  and projections \(\tilde{P}_n^i\) , \(n\in \mathbb {Z}^-\)  such that \(\operatorname{Ker}\tilde{P}_0^-=V_{i} \oplus \ldots \oplus V_r\) . Since,
\(\mathbb {R}^d=S_{i}\oplus V_{i} \oplus \ldots \oplus V_r,\) 
we conclude from the proof of [1]} that \((\frac{1}{\tilde{c}_i}B_n)_{n\in \mathbb {Z}}\)  admits a strong exponential dichotomy on \(\mathbb {Z}\)  with respect to the sequence of norms \(\mathcal {S}^{\prime }\) .
This implies that \((0, \infty )\setminus \Sigma _{ED, \mathbb {B}, \mathcal {S}} \subset (0, \infty )\setminus \Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}\) , i.e. \(\Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}\subset \Sigma _{ED, \mathbb {B}, \mathcal {S}}\) . On the other hand, we trivially have that \(\Sigma _{ED, \mathbb {B}, \mathcal {S}}\subset \Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}\) , proving that the two spectra coincide, i.e.,
\(\Sigma _{ED, \mathbb {B}^{\prime }, \mathcal {S}^{\prime }}=\Sigma _{ED, \mathbb {B}, \mathcal {S}}=\bigcup _{i=1}^r [a_i, b_i].\) 
The conclusion of the theorem now follows readily from Theorem REF  given in the Appendix A (which is a corollary of [2]}).
